Lyapunov Exponents for Infinite Dimensional

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Zeng Lian, Courant Institute KeninG Lu,
Brigham Young University

• Lyapunov Exponents for Infinite Dimensional
• Random Dynamical Systems in a Banach Space

International Conference on Random Dynamical
Systems Chern Institute of Mathematics,
Nankai, June 8-12 2009

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Content

• 1. Random Dynamical Systems
• 2. Basic Problems
• 3. Linear Random Dynamical Systems
• 4. Brief History
• 5. Main Results
• 6. Nonuniform Hyperbolicity

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1. Random Dynamical Systems

• Example 1 Quasi-period ODE
• where
  is nonlinear
• Let be
  the Haar measure on .
• Then
• is a probability space and is a
  DS preserving
• Let be the solution of
  Then

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1. Random Dynamical Systems

• Example 2. Stochastic Differential Equations
• where
• is the
  Brownian motion.

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1. Random Dynamical Systems

• The classical Wiener Space
• Open compact topology
• is the Wiener measure
• The dynamical system
• is invariant and ergodic under
• The solution operator generates RDS

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1. Random Dynamical Systems

• Example Random Partial Differential Equation
• where -Banach space and is a
  measurable
• dynamical system over probability space
• Random dynamical system solution operator

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1. Random Dynamical Systems

• Metric Dynamical System
• Let be a probability space.
  
• Let
  be a metric dynamical system
• (i)
• (ii)
• (iii) preserves the probability
  measure
• Evolution of Noise

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1. Random dynamical systems

• A map
• is called a random dynamical system over
  if
• is measurable
• the mappings
  form a
• cocycle over

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1. Random Dynamical Systems

• Time-one map
• Random map generates
• the random dynamical system

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2. Basic Problems Mathematical
Questions

• Two Fundamental Questions
• Mathematical Model
• Question 1.

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2. Basic Problems Mathematical
Questions

• Mathematical Model
• Computational Model
• Question 2
• Can we trust what we see?

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2. Basic Problems Mathematical
Questions

• Stability
• Sensitive dependence of initial data

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2. Basic Problems

• Deterministic Dynamical Systems
• Stationary solutions
• Eigenvalues
• Eigenvectors

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Random Dynamical Systems

• Deterministic Dynamical Systems
• Periodic Orbits
• Floquet exponents
• Floquet spaces

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Random Dynamical Systems

• Random Dynamical Systems
• Orbits
• Linearized Systems
• Lyapunov exponents
• measure the average rate of separation of orbits
  starting from nearby initial points.

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3. Dynamical Behavior of Linear RDS

• The Linear random dynamical system generated by
  S
• Basic Problem
• Find all Lyapunov exponents

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4. Brief History

• Finite Dimensional Dynamical Systems
• V. Oseledets, 1968 (31 pages)
• Existence of Lyapunov exponents
• Invarant subspaces,.
• Multiplicative Ergodic Theorem
• Different Proofs
• Millionshchikov Palmer, Johnson, Sell
  Margulis KingmanRaghunathan Ruelle Mane
  Crauel Ledrappier Cohen, Kesten, Newman
  Others.

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4.Brief History

• Applications
• Deterministic Dynamical Systems
• Pesin Theory, 1974, 1976, 1977
• Nonuniform hyperbolicity
• Entropy formula, chaotic dynamics
• Random Dynamical Systems
• Ruelle inequality, chaotic dynamics
• Entropy Formula, Dimension Formula
• Ruelle, Ladrappia, L-S. Young,
• Smooth conjugacy
• W. Li and K. Lu

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4. Brief History

• Infinite Dimensional RDS
• Ruelle, 1982 (Annals of Math)
• Random Dynamical Systems in a Separable Hilbert
  Space.
• Multiplicative Ergodic Theorem

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4. Brief History

• Basic Problem
• Establish Multiplicative Ergodic Theorem for RDS
• Banach space such as

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Brief History

• Infinite Dimensional RDS
• Mane, 1983
• Multiplicative Ergodic Theorem

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4. Brief History

• Infinite Dimensional RDS
• Thieullen, 1987
• Multiplicative Ergodic Theorem

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4. Brief History

• Infinite Dimensional RDS
• Flandoli and Schaumlffel, 1991
• Multiplicative Ergodic Theorem

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4. Brief History

• Infinite Dimensional RDS
• Schaumlffel, 1991
• Multiplicative Ergodic Theorem

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Main Results

• Infinite Dimensional RDS
• Lian L, Memoirs of AMS 2009
• Multiplicative Ergodic Theorem
• Difficulties
• Random Dynamical Systems
• No topological structure of the base space
  
• Banach Space
• No inner product structure

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5. Main Results

• Settings and Assumptions
• --- Separable Banach Space
• Measurable metric dynamical system
• is strongly
  measurable map

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5. Main Results

• Measure of Noncompactness
• Let
• Kuratowski measure of noncompactness
• Index of noncompactness for a map

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5. Main Results

• Measure of Noncompactness
• If S is a bounded linear operator,
• is the radius of essential specrtrum of S

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5. Main Results

• Principal Lyapunov Exponent
• Exponent of Noncompactness
• When LRDS is compact

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5. Main Results

• Theorem A (Lian L, Memoirs of AMS 2009)
• Assume that
• Then, ( ? -invariant subset of full
  measure)
• there are finitely many Lyapuniv exponents
• and invariant splitting

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5. Main Results

• such that
• (1) Invariance
• (2) Lyapunov exponents
• for all

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5. Main Results

• (3) Exponential decay rate in
• and

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5. Main Results

• Measurability
• 1. are
  measurable
• 2. All projections are strongly
  measurable
• (5) All projections are tempered

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5. Main Results

• (II) There are many Lyapuniv exponents
• and many finite dimensional subpaces
• and many infinite dimensional
  subpaces
• such that

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5. Main Results

• and
• (1) Invariance
• (2) Lyapunov exponents
• for all

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5. Main Results

• (3) Exponential decay rate in
• and

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5. Main Results

• Theorem B (Lian L, Memoirs of AMS 2009)
• Theorem A holds
• for continuous time random dynamical systems

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6. Nonuniform Hyperbolicity

• Theorem C There are ? -invariant random variable
  ?(?) gt0 and tempered random variable K(?) 1
  such that
• where are the
  stable, unstable projections

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7. Random Stable and Unstable Manifolds

• Theorem D (Lian L, Memoirs of AMS 2009)

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7. Random Stable and Unstable Manifolds
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8. Application
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8. Application
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8. Application
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9. Outline of Proof

• Volume function
• Kingmans additive erogidc theorem
• Katos space gap
• Measurable selection theorem
• Measurable Hahn-Banach theorem
• Measure theory

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